Cut-model inference [1]} has proven very effective in this setting. We discuss this in detail below. It can be thought of as a kind of sequential imputation procedure, in which the distribution of a shared parameter is imputed form the information in one module and then passed on as a kind of prior for the shared parameter in a second module. This is not Bayesian inference, as information from the second module does not inform the shared parameter. A early form of Cut model inference has been available in WinBUGS [2]} for some time.
Cut models have found many applications: air pollution [3]}, epidemiological models [4]}, [5]}, [6]}, [7]}, [8]}, meta-analysis [9]}, [10]}, [11]} and propensity scores [12]}, [13]}. [14]} gives an overview of modularized Bayesian analysis including Cut-models from the perspective of statistical decision theory and [15]} gives asymptotic properties. Nested MCMC [1]} is commonly used for fitting Cut models. New developments include variational approximation [17]} and a computationally efficient variant of nested-MCMC [18]}.
